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Stabilization and Disturbance Rejection for the Beam Equation
Ömer Morgül
Abstract—We consider a system described by the Euler–Bernoulli beam
equation. For stabilization, we propose a dynamic boundary controller ap-plied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem.
Index Terms—Boundary control systems, distributed parameter systems,
disturbance rejection, flexible structures, semigroup theory, stability.
I. INTRODUCTION
Many mechanical systems, such as spacecraft with flexible attach-ments, or robots with flexible links, and many practical systems such as power systems, mass transport systems contain certain parts whose dy-namic behavior can be rigorously described only by partial differential equations (PDEs). In such systems, to achieve high precision demands, the dynamic effect of the system parts whose behavior are described by PDEs on the overall system has to be taken into account in designing the controllers.
In recent years, boundary control of systems represented by PDEs has become an important research area. This idea is first applied to the systems represented by the wave equation (e.g., elastic strings, cables), see, e.g., [2], [8], and then extended to the beam equations, [3], and to the rotating flexible structures, see [11], [12]. In particular, it has been shown that for a beam which is clamped at one end and is free
Manuscript received May 26, 2000; revised January 22, 2001. Recommended by Associate Editor I. Lasiecka.
The author is with the Department of Electrical and Electronics, En-gineering, Bilkent University, 06533 Bilkent, Ankara, Turkey (e-mail: morgul@bilkent.edu.tr).
Publisher Item Identifier S 0018-9286(01)11109-8.
at the other end, a single nondynamic boundary control applied at the free end is sufficient to exponentially stabilize the system, see [3]. This result was then extended for dynamic boundary controllers, see [13]. For more references and technical information on the subject the reader is referred to [10].
In this note, we consider a linear time invariant system which is represented by one-dimensional Euler–Bernoulli beam equation in a bounded domain. We assume that the system is clamped at one end and the boundary control input is applied at the other end. For this system, we propose a finite dimensional dynamic boundary controller. This in-troduces extra degrees of freedom in designing controllers which could be exploited in solving a variety of control problems, such as distur-bance rejection, pole assignment, etc., while maintaining stability. The transfer function of the controller is a proper rational function of the complex variables, and may contain a single pole at s = 0 and an-other ones = j!1,!1 6= 0, provided that the residues corresponding
to these poles are nonnegative; the rest of the transfer function is re-quired to be a strictly positive real function. Such transfer functions are called marginally stable positive-real (MSPR) functions, see [7]. This type of controllers have been proposed before for the stabilization of wave equation, see [14] for the stabilization (only simple pole ats = 0 is used), and [15] for disturbance rejection. While an exponential sta-bility result has been given in [14], only asymptotic stasta-bility result has been given in [15]. A similar controller without any pole on the imag-inary axis has been proposed for the beam equation in [13]. We then show that if!1does not belong to a countable set (e.g., the zeros of a transcendental function), then the closed loop system is asymptotically stable in general, and is exponentially stable in some special cases. We note that in many cases exponential stability is desired, due to, e.g., the robustness of the resulting closed-loop system, and in infinite di-mensional systems, asymptotic stability may not imply exponential sta-bility. We also relate this set of zeroes with the transmission zeroes of the appropriate transfer function. We then consider the case where the output of the controller is corrupted by disturbance. We show that if the structure of the disturbance is known (i.e., the frequency spectrum), then it may be possible to choose the controller accordingly to atten-uate the effect of the disturbance at the system output.
This paper is organized as follows. In Section II, we introduce the system considered and propose a class of controller for stabilization. In Section III, we give some stability results. In Section IV, we con-sider disturbance rejection problem. Finally, we give some concluding remarks in Section V.
II. PROBLEMSTATEMENT
We consider a flexible beam clamped at one end and is free at the other end. Without loss of generality, we assume that the beam length, mass density and the flexural rigidity are given asL = 1, = 1 andT = 1, respectively. We denote the displacement of the beam by u(x; t) at x 2 (0; 1) and t 0. The beam is clamped at one end and is controlled by a boundary control force at the other end. The equations are given as (x 2 (0; 1), t 0)
utt+ uxxxx= 0; (1)
u(0; t) = 0; ux(0; t) = 0 (2)
uxx(1; t) = 0; uxxx(1; t) = f(t) (3) where a subscript, as inutdenotes a partial differential with respect to the corresponding variable, andf(1): R+! R is the boundary control
force applied at the free end of the beam. 0018–9286/01$10.00 © 2001 IEEE
In this paper we assume thatf(t) is generated by a dynamic con-troller whose relation between its inputut(1; t), and its output f(t) is
given by the following: ^
f(s) = g(s)^ut(1; s) (4)
where a hat denotes the Laplace transform of the corresponding vari-able. In (4),g(s) is the controller transfer function, which is assumed to be a proper and rational function ofs. We assume that the controller transfer functiong(s) has the following form:
g(s) = g1(s) + ks + ks2+ !1s 2 1;
(5) whereg1(s) is a strictly positive-real function, and k; k1; !12 R.
In [3], a static controller withg(s) = d > 0 was considered and it was shown that the closed-loop system is exponentially stable. The dynamic controller given by (5) withk = k1 = 0 was considered in
[13] for the beam equation and an exponential stability result was given. For the wave equation, in [14] the casek1= 0 was considered and an
exponential stability result was given, and in [15] only an asymptotic stability result was given for the casek1> 0.
To analyze the well-posedness of the system given by (1)–(3), (4), we need a state-space representation for the controller given by (4) and (5). Let(A; b; c; d) be a minimal (i.e., controllable and observable) rep-resentation ofg1(s). Then, noting that the term k actually corresponds
to an integrator, we obtain the following state-space representation for the controller given by (4), (5):
_z1= Az1+ but(1; t) (6)
_x1= !1x2; _x2= 0!1x1+ ut(1; t) (7) f(t) = cTz
1+ dut(1; t) + ku(1; t) + k1x2 (8) whereg1(s) = cT(sI 0 A)01b + d, z1 2 Rn, for some natural number n, is the actuator state, A 2 Rn2n is a constant matrix, b; c 2 Rnare constant column vectors,d 2 R, and the superscript
T denotes transpose.
We make the following assumptions concerning the actuator given by (6), (8) throughout this work.
Assumption 1: All eigenvalues ofA 2 Rn2nhave negative real parts.
Assumption 2: (A; b) is controllable and (c; A) is observable. Assumption 3: d 0; k 0; k1 0; moreover there exists a
constant , d 0, such that the following holds:
d + Re cT(j!I 0 A)01b > ; ! 2 R (9) whereg1(s) = cT(sI 0 A)01b + d. Moreover for d > 0, we assume
> 0 as well.
III. STABILITYRESULTS
Let the Assumptions 1–3 stated above hold. Then, since the transfer function g1(s) = d + cT(sI 0 A)01b is strictly positive real it
follows from the Meyer–Kalman–Yakubovich Lemma that given any symmetric positive definite matrixQ 2 Rn2n, there exists a symmetric positive definite matrixP 2 Rn2n, a vectorq 2 Rnand a constant > 0 satisfying (see [19, p. 133])
ATP + P A = 0qqT 0 Q (10)
P b 0 c = 2(d 0 ) q: (11)
To analyze the system given by (1)–(3), (6)–(8), we first define the function spaceH as follows:
H = (uvz1x1x2)T u 2 H20; v 2 L2;
z12 Rn; x1; x22 R; (12) where the spacesL2, andHk0 are defined as follows:
L2= f: [0; 1] ! R 1 0 f 2dx < 1 ; (13) Hk= f 2 L2 f0; . . . ; f(k)2 L2 ; Hk 0 = f 2 Hk f(0) = f0(0) = 0 : (14)
The equations (1)–(3), (6)–(8) can be written in the following ab-stract form:
_z = Lz; z(0) 2 H (15)
wherez = (uutz1x1x2)T 2 H, the operator L: H ! H is a linear
unbounded operator defined as
L u v z1 x1 x2 = v 0uxxxx Az1+ bv(1) !1x2 0!1x1+ v(1) : (16)
The domainD(L) of the operator L is defined as: D(L) = (uvz1x1x2)T 2 H u 2 H40;
v 2 H2
0; z12 Rn; x1; x22 R;
uxx(1) = 0; 0uxxx(1) + cTz1+ dv(1)
+ ku(1) + k1x2= 0 : (17)
Let the Assumptions 1–3 hold, letQ 2 Rn2nbe an arbitrary sym-metric positive–definite matrix and letP 2 Rn2n,q 2 Rn be the solutions of (10) and (11) whereP is also a symmetric and positive definite matrix. InH, we define the following “energy” inner-product: hy; ~yiE =12 1 0 v~v dx + 1 2 1 0 uxx~uxxdx + 1 2ku(1)~u(1) +1 2z1TP z1+12k1(x1~x1+ x2~x2) (18)
wherey = (uvz1x1x2)T, ~y = (~u~v~z1~x1~x2)T 2 H. It can be shown
thatH, together with the energy inner-product given by (18) becomes a Hilbert space. The “energy” norm induced by (18) [for the solution z(t) of (15)] is given by: E(t) = kz(t)k2= 1 2 1 0 u 2 tdx +12 1 0 u 2 xxdx +1 2ku2(1; t) +12zT1P z1+12k1(x21+ x22): (19)
Theorem 1: Consider the system given by (15) withd 0, k 0 andk1 0.
i) The operatorL generates a C0-semigroup of contractionsT (t) in H, (for the terminology of semigroup theory, the reader is referred to, e.g., [17]).
ii) Ifk1 > 0 and ! = !1is not one of the roots of the following transcendental equation:
then the semigroupT (t) generated by L is asymptotically stable, that is all solutions of (15) asymptotically converge to zero. Proof: i) We use Lumer–Phillips theorem, to prove the assertion i), see, e.g., [17]. To prove thatL is dissipative, we compute hy; LyiEby using (16) in (18). Then, integrating by parts, and using (10), (11), (14), (17) we obtain hy; LyiE= 0 2v2(1) 0 14 2(d 0 ) v(1) 0 zT 1q 2 0 4zT 1Qz1: (21)
Sincehy; LyiE 0, it follows that L is dissipative, (see [14] for similar calculations).
After some straightforward calculations, it can be easily shown that I 0 L: H ! H is onto for > 0, (see [14] for similar calculations). Then, it follows from the Lumer–Phillips theorem thatL generates a C0-semigroup of contractionsT (t) on H.
ii) To prove the assertion ii), we use LaSalle’s invariance principle, extended to infinite dimensional systems, see [18] and [10]. According to this principle, all solutions of (15) asymptotically tend to the max-imal invariant subset of the following set:
S = fz 2 Hj _E = 0g (22)
provided that the solution trajectories fort 0 are precompact in H. Since the operatorL: H ! H generates a C0-semigroup of contrac-tions onH (hence, the solution trajectories are bounded on H for t 0), the precompactness of the solution trajectories are guaranteed if the operator(I 0L)01:H ! H is compact for some > 0, see [10]. To prove the last property, we first show thatL01exists and is a compact operator onH. To see this, let q = (fhrr1r2)T 2 H be given. We want
to solve the equationLz = q for z, where z = (uvz1x1x2)T 2 D(L). The solution of this equation can easily be found as:
u(x) = 0 x 0 0 0 0 h() d d1d2d3 + c1x3+ c2x2; v(x) = f(x) (23) z1= A01(r 0 f(1)b); x1= f(1) 0 r! 2 1 ; x2= r 1 !1 (24)
where the constantsc1; c2can be uniquely determined from (8). It fol-lows thatL01exists and mapsH into H4 2 H22 Rn2 R 2 R, moreover(uvz1x1x2)T 2 D(L). Since q = (fhrr1r2)T 2 H it follows thatf 2 H20, see (12). Hence, ifkqk is bounded in H, it fol-lows easily that thatf(1) is bounded as well. Therefore L01maps the bounded sets ofH into the bounded sets of H42 H22 Rn2 R 2 R. Since the embedding of the latter intoH is compact, see [20, p. 14], it follows thatL01is a compact operator. This also proves that the spec-trum ofL consists entirely of isolated eigenvalues, and that for any in the resolvent set ofL, the operator (I 0 L)01:H ! H is a com-pact operator, see [9, p. 187]. Furthermore, our argument above shows that = 0 is not an eigenvalue of L. Since the operator L generates aC0-semigroup of contractions onH, by the argument given above it follows that the solutions trajectories of (15) are precompact inH for t 0, hence by LaSalle’s invariance principle, the solutions asymp-totically tend to the maximal invariant subset ofS [see (22)]. Hence, to prove that all solutions of (15) asymptotically tend to the zero solution, it suffices to show thatS contains only the zero solution, which is a typical procedure in the application of LaSalle’s invariance principle.
By using (15) and (19), we see that _E = 2hz; LziE. Hence, from (21) we obtain _E = 2hz; LziE= 0 u2t(1; t) 0 12 2(d 0 )ut(1; t) 0 zT1q 2 0 2zT 1Qz1: (25)
To prove thatS contains only the zero solution, we set _E = 0 in (25), which results inz1 = 0. This implies that _z1= 0, hence by using (6)
and (8) we obtainut(1; t) = 0, f(t) = ku(1; t) + k1x2. Hence, all solutions of (15) inS satisfy the following:
utt+ uxxxx= 0; _x1= !1x2; _x2 = 0!1x1 (26) u(0; t) = 0; ux(0; t) = 0;
uxx(1; t) = 0; ut(1; t) = 0 (27) uxxx(1; t) = ku(1; t) + k1x2: (28)
Consider the system given by (26), (27). This system can be put into the form _zp = Lpzpwithzp= (uutx1x2)T 2 Hp, whereHpis the same as given by (12) with obvious omission ofz1, andLpis similar toL given by (16) with obvious omission of z1(i.e., the third row) and v(1) = 0. D(Lp) is given as (17) with the omission of z1, and the
last boundary condition should be replaced byv(1) = 0. Hpis also a Hilbert space with the inner-producth1; 1ipinduced by (18) with the omission ofz1. By straightforward calculation, and using integration by parts, it can easily be shown thathzp; Lp~zpip = 0hLpzp; ~zpip, for anyzp; ~zp 2 D(Lp), hence from [5, Th. 4.1], it follows that Lp
is skew-adjoint, i.e.,Bp= jLpis self-adjoint. Hence, it follows from [17, Th. 10.8] thatLpgenerates aC0semigroup of contractions. Also, by using [5, Th. 5.1], it follows that there exists a complete set of or-thonormal basisf'1; '2; . . .g on Hpconsisting of eigenvectors ofLp. Moreover, for anyz0 2 D(Lp), we have z0 = jhz0; 'jip'j and Lpz0= jjhz0; 'jip'j, wherejdenotes the eigenvalues ofLp, see [5, Th. 5.1]. Hence, the solutionzp(t) of (26), (27) can be given
aszp(t) = jcje t'j, where the coefficientscj 2 C can be
de-termined from initial conditions and the eigenvectors' have the form ' = (uvx1x2)T 2 D(Lp). Due to the structure of Lp, = 6j!1is an eigenvalue pair, and since =p!1is not a root of (20), the cor-responding eigenvectors haveu = v = 0. The rest of the eigenvalues have the form = 6j 2, where is a root of (20), and since we have 6= p!1, the corresponding eigenvectors havex1 = x2 = 0 with
u(0) = u0(0) = u00(1) = v(1) = 0. By using these in (28), after
some straightforward calculations, we obtaincj = 0, j = 1; 2; . . ..
Hence,zp = 0 is the only possible solution of (26)–(28). Hence, by
LaSalle’s invariance theorem, we conclude that the solutions of (15) asymptotically tend to the zero solution.
Theorem 1 remains valid even ifk1= 0, provided that the variables x1andx2are suppressed everywhere. The proof of this fact is essen-tially the same as the proof of Theorem 1.
It was proven in [13] that for k = k1 = 0, if d > 0, then the
closed-loop system (1)–(3), (6) and (8) is exponentially stable. Since the subsystem (7) is essentially finite dimensional, we may expect the same conclusions hold for the casek 0; k1> 0 as well. In the sequel
we will prove this result by using Huang’s Theorem stated below: Theorem (Huang): LetL be a linear operator on a Hilbert space H. Assume thatL generates a bounded C0semigroupT (t) on H. Then, T (t) is exponentially stable if and only if the following holds:
i) imaginary axis belongs to the resolvent set ofL. ii) the following resolvent estimate holds:
sup
!2Rk(j!I 0 L)
01k < 1: (29)
Proof: See [6] or [10] for an alternative proof.
In the sequel, we will work on the complexified versions of the Hilbert spaces mentioned above; for convenience we do not change the notation.
Theorem 2: Consider the system given by (15) withd > 0, k 0 andk1 > 0. Let the assumptions 1–3 hold. If ! = !1is not a root of (20), then the semigroupT (t) generated by L is exponentially stable.
Proof: We will use Huang’s theorem cited above. First note that by Theorem 1,T (t) is bounded. If = p!1 is not a root of (20), then by the part ii) of Theorem 1, the semigroupT (t) generated by L is asymptotically stable, hence L cannot have an eigenvalue on the
imaginary axis, and sinceL has compact resolvent, it follows that the imaginary axis belongs to the resolvent set ofL, see, e.g., [10, Th. 3.26]. Next, we will show that the resolvent estimate given by (29) holds for sufficiently large!. The calculations are lengthy, but straightforward, similar to the ones given in [4]. Here we will omit the details, and give only basic steps of the calculations. Lety = (pqrr1r2)T 2 H and , in the resolvent set ofL, be given and let z = (uvz1x1x2)T 2 D(L)
be the solution of the following:
(I 0 L)z = y: (30)
In the sequel, we will use = j!, ! 2 R, for simplicity. The solution u of (30) satisfying u(0) = ux(0) = 0 is given by
u(x) = A(cosh x 0 cos x) + B(sinh x 0 sin x) + 12 3 x
0 [sinh (x 0 ) 0 sin (x 0 )]
1 [p() + q()] d; =p! (31)
whereA and B are determined by the remaining boundary conditions uxx(1) = 0; uxxx(1) 0 g()u(1) = k0 g() p(1)
+k1r220 !+ !12r1 1 + c
T(I 0 A)01r (32)
andg(1) is given by (5). Using (31) in (32) we obtain A = 11 cosh + cos 0 g() 3 (sinh 0 sin )
1X10 (sinh + sin )(X2+ X3+ X4) (33)
B = 11 0 sinh + sin + g() 3 (cosh 0 cos ) 1X1+ (cosh + cos )(X2+ X3+ X4) (34) where
1 = 2 1 + cosh cos 0 g() 3
(sinh cos 0 cosh sin ) (35)
X1= 0 12 3 1 0 [sinh (1 0 ) + sin (1 0 )] 1 [p() + q()] d (36) X2= 1 3 k0 g() p(1) + k 13 r220 !+ !12r1 1 + cT(I 0 A) 3 01r (37) X3= 0 12 3 1 0 [cosh (1 0 ) + cos (1 0 )] 1 [p() + q()] d (38) X4= 12 3 g() 3 1 0 [sinh (1 0 ) 0 sin (1 0 )] 1 [p() + q()] d: (39)
(Note that1 = 0 may be considered as the characteristic equation in the sense that its roots are the eigenvalues of the closed-loop system.) In the sequel, by using estimates of (33)–(39), we will show that an estimate of the formkzk Mkyk holds. We will use the following notation: for any functionF (1) > 0, O(F ( )) denotes any function
which satisfiesO(F ( )) MF ( ) for some M > 0 and for suffi-ciently large. Also, for anyf 2 L2,kfk2denotes theL2norm off.
For simplicity, assume that > 0. Using integration by parts in (36), (38) and (39), we obtain Xi= 0 14 3e 1 0 e 0 [jp xx() + q()] d + O kpxxk2 + kqk3 2 ; i = 1; 3 (40) X4= jg() 3 p(1) + g()4 6 e 1 0 e 0 [jp xx() + q()] d + O kpxxk2 + kqk4 2 : (41)
Let^u denote the integral term in (31). Upon differentiating and using integration by parts, we obtain
^uxx(x) = 14 e x 1 0 e 0 [jp xx() + q()] d +O kpxxk2 + kqk2 : (42) In (35), by collecting dominant terms, we obtain
1 = 2 + e cos 0 jg()
(cos 0 sin ) + O(e0 ): (43) Letg() = R(!)+jI(!)where R and I denote the real and imaginary parts, respectively. Moreover we haveR(!) > 0 and I(!) = O(1= 2) for large . Noting that cos2 + (cos 0 sin )2 c > 0
for somec, from (43) we obtain 1 = O(e = ) for large .
By using (33), (34), (40), (41) and (42) in (31) and noting that 101 = O( e0 ), after straightforward algebraic calculations we
obtain the following estimate for sufficiently large:
kuxxk2 M1kyk (44)
whereM1 > 0 is a constant and the norm k 1 k is given by (19). Note
that in this calculation the key point is the fact that the dominant terms in (40)–(42), which are the integral terms, cancel in the expression of uxx, see [4] for a similar result. By using (33), (34), (40), (41), (42) and (31) inv = u 0 p, [see (30)], after straightforward calculations similar to the ones mentioned above we obtain the following estimate for sufficiently large:
kvk2 M2kyk (45)
whereM2 > 0 is a constant. Similarly, from (30) we obtain kz1kn= O jv(1)j + krk 2 n
jxij = O jv(1)j + jr 21j + jr2j ; i = 1; 2 (46) wherek1kndenotes the norm inRn, see (19). Finally. note thatv(1) = u(1) 0 p(1), and ju(1)j kuxxk2,jp(1)j kpxxk2. Combining these, we obtain the following inequality for sufficiently large :
kzk Mkyk (47)
whereM > 0 is a constant, and for a given y 2 H, z 2 D(L) is the solution of (30). Hence for some > 0 sufficiently large we have
sup
!k(j!I 0 L)
SinceL has compact resolvent and the imaginary axis is in the resolvent set, we also have the following:
sup
!k(j!I 0 L)
01k < 1: (49)
Combining (48) and (49) we obtain (29). Hence, by Huang’s Theorem, L generates an exponentially stable semigroup in H.
Theorem 2 remains valid even ifk1= 0, provided that the variables
x1andx2are suppressed everywhere. The proof of this fact is essen-tially the same as the proof of Theorem 2.
IV. DISTURBANCEREJECTION
In this section, we show the effect of the proposed control law given by (6)–(8) on the solutions of the system given by (1)–(3), when the output of the controller is corrupted by a disturbancen(t), that is (8) has the following form:
f(t) = cTz
1+ dut(1; t) + ku(1; t) + k1x2+ n(t) (50)
or, equivalently, (4) has the following form: ^
f(s) = g(s)^yt(1; s) + ^n(s) (51) where^n(s) is the Laplace transform of the disturbance n(t) and g(s) is given by (5).
To find the transfer function fromn(t) to ut(1; t), first we need to
find the transfer function fromf(t) to ut(1; t). By taking the Laplace transform of (1)–(3) and using zero initial conditions, after some straightforward calculations we obtain the following:
^ut(1; s) = 0h(s) ^f(s);
h(s) = j (cosh sin 0 sinh cos ) (1 + cosh cos ) (52) wheres2= 0 4, see also [1]. By using (52) in (51), we obtain
^ut(1; s) = 01 + h(s)g(s)h(s) ^n(s): (53)
Remark 1: Consider the system given by (1)–(3). If we considerf as an input andut(1; t) as output, it is known that this system is pas-sive, see [10]; moreover the transfer function of this system is given by h(s) in (52). On the other hand, the controller transfer function g(s) which is given by (5) is MSPR, see [7]. It was shown in [7] that for finite-dimensional linear, time-invariant systems (LTI), in the classical negative feedback configuration ifh(s) (the system to be controlled) is positive real and ifg(s) (the controller) is MSPR, then the closed-loop system is asymptotically stable, provided that none of the imaginary axis poles of g(s) is a transmission zero of h(s). Note that this re-sult may not hold for infinite dimensional systems. In our case, (1)–(3) represent a passive system, which is equivalent to positive realness in finite dimensional LTI systems. It can easily be shown that! = !1
is a root of (20) if and only ifh(j!1) = 0. In this sense, Theorem 2
may be considered as an extension of the stated result of [7] to an infi-nite dimensional system given by (1)–(3). Note that, although in fiinfi-nite dimensional LTI case the asymptotic stability implies exponential sta-bility, this is not necessarily true in infinite dimensional systems. We also note that a similar result holds for the wave equation, [16].
Remark 2: The controller given by (5) can easily be generalized to
g(s) = g1(s) + ks+ N i=1 kis s2+ !2 i (54)
for anyN. Theorem 2 will remain valid, provided that ki 0 and !i
is not a root of (20) fori = 1; 2; . . . ; N.
From (53), we can also derive a procedure to designg(s) if we know the structure ofn(t). For example if n(t) has a band-limited frequency spectrum, (i.e., has frequency components in an interval of frequencies [1; 2]), then we can chooseg(s) to minimize
c(!) = 1 + h(j!)g(j!)h(j!) ; ! 2 [1; 2] : (55) Note that to ensure the stability of the closed-loop system,g1(s) should
be a strictly positive real function as well, [see (5)]. As a simple ex-ample, assume thatn(t) = a cos !0(t). Then we may choose g(s) in
the form (5) with!1 = !0. Provided that the Assumptions 1–3 are satisfied and thatj!0is not a zero ofh(s), the closed-loop system is asymptotically stable, (see Theorem 1). Moreover, ifk1> 0, then c(!)
given above satisfiesc(!0) = 0. From (55), we may conclude that this
eliminates the effect of the disturbance at the outputut(1; t).
V. CONCLUSION
In this note, we considered a linear time invariant system which is represented by one-dimensional Euler–Bernoulli beam equation in a bounded domain. We assumed that the system is clamped at one end and the boundary control force input is applied at the other end. For this system, we proposed a finite dimensional dynamic boundary controller. This introduces extra degrees of freedom in designing con-trollers which could be exploited in solving a variety of control prob-lems, such as disturbance rejection, pole assignment, etc., while main-taining stability. The transfer function of the controller is a proper ra-tional function of the complex variables, and may contain a single pole ats = 0 and another one s = j!1,!1 6= 0, provided that the
residues corresponding to these poles are nonnegative; the rest of the transfer function is required to be a strictly positive real function. We then proved that the closed-loop system is asymptotically stable pro-vided thats = j!1 is not a zero of an appropriate system transfer function, and is exponentially stable in some cases. We also studied the case where the output of the controller is corrupted by a disturbance. We showed that, if the frequency spectrum of the controller is known, then by choosing the controller appropriately we can obtain better dis-turbance rejection.
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Robust Control of Discrete-Time Markovian Jump Linear Systems With Mode-Dependent Time-Delays
E. K. Boukas and Z. K. Liu
Abstract—This note considers the class of discrete-time Markovian jump
linear system with norm-bounded uncertainties and time-delay, which is dependent on the system mode. Linear matrix inequality (LMI) -based suf-ficient conditions for the stability, stabilization and control are de-veloped. A numerical example is worked out to show the usefulness of the theoretical results.
Index Terms—Discrete-time Markovian jump linear system, con-trol, linear matrix inequality (LMI), time-delay system.
I. INTRODUCTION
Discrete-time Markovian jump linear system is a hybrid one with state comprised of two components: a discrete part denoted byrtand a continuous part, denoted byxt. Discrete statert is a discrete-time Markov chain representing the mode of the system andxt denotes the physical state of the system, e.g., the inventory level in manufac-turing systems. The continuous statextevolves according to a differ-ence equation when the mode is fixed. For more information on dis-crete-time Markovian jump linear systems, the reader is referred to [5], [6], and the references therein.
Time-delay occurs frequently in many practical systems, such as manufacturing system, telecommunication and economic systems etc., which is an important source of instability and poor performance. For continuous-time Markovian jump linear systems with time-delay, we refer the reader to [4]. For discrete-time Markovian jump linear system with time-delay, [1] studied the robust stability, stabilization andH1
problem. The purpose of this note is to extend the results in [1] to the
Manuscript received January 19, 2001; revised June 11, 2001. Recommended by Associate Editor Q. Zhang.
The authors are with the Mechanical Engineering Department, École Poly-technique de Montréal, Montréal, QC, H3C 3A7 Canada.
Publisher Item Identifier S 0018-9286(01)11108-6.
case where the time-delay in the system is dependent on the system mode.
The goal of this note is to study the robust stability and robust stabi-lizability of the class of discrete-time Markovian jump linear systems with time-delay and norm bounded uncertainties. The robustH1 con-trol is also considered. The sufficient conditions we will establish are all in linear matrix inequality (LMI) formalism which makes their res-olution easy. The rest of this note is organized as follows: Section II describes the system model. Section III addresses the robust stability and stabilization problem. Section IV studies the robustH1control problem. Section V provides a numerical example to show the valid-ness of the proposed results.
II. MODELDESCRIPTION
Let frk; k 0g be a Markov chain with state space
S = f1; . . . ; Ng and state transition matrix P = [pij]i; j2S,
i.e., the transition probabilities offrk; k 0g are as follows:
P [rk+1= jjrk= i] = pij; 8 i; j 2 S
withpij 0, 8 i; i 2 S and Nj=1pij = 1, for i 2 S.
Consider a discrete-time hybrid system withN modes. Suppose that the system mode switching is governed byfrk; k 0g and the system
parameters contain norm-bounded uncertainties. Let the system dy-namics be described by the following:
xt+1= A(t; rt)xt+ Ad(t; rt)xt0 (r ) +B(rt; t)ut+ B1(rt)wt; xs= s; s = 0 ; . . . ; 01; zt= C(rt; t)xt+ Cd(rt; t)xt0 (r ) +Bc(rt; t)u(t) + C1(rt)wt (1)
wherext2 nis the state of the system, for eachrt2 S
A(rt; t) = A(rt) + 1a(rt; t) Ad(rt; t) = Ad(rt) + 1d(rt; t) B(rt; t) = B(rt) + 1b(rt; t) C(rt; t) = C(rt) + 1c(rt; t) Cd(rt; t) = Cd(rt) + 1cd(rt; t) Bc(rt; t) = Bc(rt) + 1bc(rt; t)
with A(rt), Ad(rt), B1(rt), B(rt), C(rt), Cd(rt), Bc(rt) and
C1(rt) are matrices with appropriate dimensions, 1a(rt; t),
1d(rt; t), 1a(rt; t), 1b(rt; t), 1c(rt; t), 1cd(rt; t), 1bc(rt; t)
are unknown matrices denoting the uncertainties in the system. (rt)
is a constant, denoting the time-delay of the system when the system is in modert.
In this note, we assume that the admissible uncertainties satisfy the following: 1a(rt; t) 1d(rt; t) 1b(rt; t) 1c(rt; t) 1cd(rt; t) 1bc(rt; t) = G1(rt) G2(rt) 1(rt; t) ( H1(rt) H2(rt) H3(rt) )
with1>(rt; t)1(rt; t) I, 8 rt 2 S. In the sequel, notation X >
0(0), with X being a matrix, means that X is symmetric and (semi-) positive–definite.
